DM204 , 2011 SCHEDULING, TIMETABLING AND ROUTING Lecture 1 Introduction to Scheduling: Terminology and Classification Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark
Course Introduction Scheduling Outline Complexity Hierarchy 1. Course Introduction 2. Scheduling Definitions Classification Exercises Schedules 3. Complexity Hierarchy .::. 2
Course Introduction Scheduling Outline Complexity Hierarchy 1. Course Introduction 2. Scheduling Definitions Classification Exercises Schedules 3. Complexity Hierarchy .::. 3
Course Introduction Scheduling Course Content Complexity Hierarchy General Optimization Methods Scheduling (Manufacturing) Mathematical Programming Single and Parallel Machine Models Flow Shops and Flexible Flow Shops Constraint Programming Job Shops Heuristics Resource-Constrained Project Scheduling Problem Specific Algorithms (Dynamic Programming, Timetabling (Services) Branch and Bound, ...) Interval Scheduling, Reservations Educational Timetabling Crew, Workforce and Employee Timetabling Transportation Timetabling Vehicle Routing Capacited Vehicle Routing Vehicle Routing with Time Windows Rich Models .::. 4
Course Introduction Scheduling Course Presentation Complexity Hierarchy Lecture plan and Schedule Communication tools Course Public Web Site (WS) ⇔ Blackboard (Bb) (public web site: http://www.imada.sdu.dk/~marco/DM204/ ) Announcements (Bb) Discussion board or Blog (Bb) not monitored Personal email (Bb) My office in working hours .::. 6
Course Introduction Scheduling Course Presentation Complexity Hierarchy Final Assessment (5 ECTS) Oral exam: 30 minutes + 5 minutes defense project meant to assess the base knowledge � based on a Case Portfolio Schedule: Oral exam: in June, day to define .::. 7
Course Introduction Scheduling Course Material Complexity Hierarchy Literature B1 Pinedo, M. Scheduling: Theory, Algorithms, and Systems Springer New York, 2008 available online B2 Pinedo, M. Planning and Scheduling in Manufacturing and Services Springer Verlag, 2005 available online B3 Toth, P. & Vigo, D. (ed.) The Vehicle Routing Problem SIAM Monographs on Discrete Mathematics and Applications, 2002 photocopies Articles and photocopies available from the web site Lecture slides .::. 8
Course Introduction Scheduling Course Goals Complexity Hierarchy How to Tackle Real-life Optimization Problems: Formulate (mathematically) the problem Model the problem and recognize possible similar problems Search in the literature (or in the Internet) for: complexity results (is the problem NP -hard?) solution algorithms for original problem solution algorithms for simplified problem Design solution algorithms and implement them Test experimentally with the goals of: checking computational feasibility configuring comparing Key ideas: Decompose problems Hybridize methods .::. 9
Course Introduction Scheduling The Problem Solving Cycle Complexity Hierarchy The real problem Modelling Experimental Analysis Mathematical Model Quick Solution: Heuristics Algorithm Implementation Mathematical Design of Theory good Solution Algorithms .::. 10
Definitions Course Introduction Classification Scheduling Exercises Outline Complexity Hierarchy Schedules 1. Course Introduction 2. Scheduling Definitions Classification Exercises Schedules 3. Complexity Hierarchy .::. 11
Definitions Course Introduction Classification Scheduling Exercises Scheduling Complexity Hierarchy Schedules Manufacturing Project planning Single, parallel machine and job shop systems Flexible assembly systems Automated material handling (conveyor system) Lot sizing Supply chain planning Services personnel/workforce scheduling public transports ⇒ different models and algorithms .::. 13
Definitions Course Introduction Classification Scheduling Exercises Problem Definition Complexity Hierarchy Schedules Constraints Activities Resources Objectives Problem Definition Given: a set of jobs J = { J 1 , . . . , J n } to be processed by a set of machines M = { M 1 , . . . , M m } . Task: Find a schedule, that is, a mapping of jobs to machines and processing times, that satisfies some constraints and is optimal w.r.t. some criteria. Notation: n , j , k jobs m , i , h machines .::. 14
Definitions Course Introduction Classification Scheduling Exercises Visualization Complexity Hierarchy Schedules Scheduling are represented by Gantt charts (a) machine-oriented (b) job-oriented .::. 15
Definitions Course Introduction Classification Scheduling Exercises Data Associated to Jobs Complexity Hierarchy Schedules Processing time p ij Release date r j Due date d j (called deadline, if strict) Weight w j Cost function h j ( t ) measures cost of completing J j at t A job J j may also consist of a number n j of operations O j 1 , O j 2 , . . . , O jn j and data for each operation. A set of machines µ jl ⊆ M associated to each operation | µ jl | = 1 dedicated machines µ jl = M parallel machines µ jl ⊆ M multipurpose machines Data that depend on the schedule Starting times S ij Completion time C ij , C j .::. 17
Definitions Course Introduction Classification Scheduling Exercises Problem Classification Complexity Hierarchy Schedules A scheduling problem is described by a triplet α | β | γ . α machine environment (one or two entries) β job characteristics (none or multiple entry) γ objective to be minimized (one entry) [R.L. Graham, E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan (1979): Optimization and approximation in deterministic sequencing and scheduling: a survey, Ann. Discrete Math. 4, 287-326.] .::. 19
Definitions Course Introduction Classification Scheduling Exercises α | β | γ Classification Scheme Complexity Hierarchy Schedules Machine Environment α 1 α 2 | β 1 . . . β 13 | γ single machine/multi-machine ( α 1 = α 2 = 1 or α 2 = m ) parallel machines: identical ( α 1 = P ), uniform p j / v i ( α 1 = Q ), unrelated p j / v ij ( α 1 = R ) multi operations models: Flow Shop ( α 1 = F ), Open Shop ( α 1 = O ), Job Shop ( α 1 = J ), Mixed (or Group) Shop ( α 1 = X ), Multi-processor task sched. Single Machine Flexible Flow Shop Open, Job, Mixed Shop ( α = FFc ) .::. 20
Definitions Course Introduction Classification Scheduling Exercises α | β | γ Classification Scheme Complexity Hierarchy Schedules Job Characteristics α 1 α 2 | β 1 . . . β 13 | γ β 1 = prmp presence of preemption (resume or repeat) β 2 precedence constraints between jobs acyclic digraph G = ( V , A ) β 2 = prec if G is arbitrary β 2 = { chains , intree , outtree , tree , sp - graph } β 3 = r j presence of release dates β 4 = p j = p preprocessing times are equal ( β 5 = d j presence of deadlines) β 6 = { s - batch , p - batch } batching problem β 7 = { s jk , s jik } sequence dependent setup times .::. 21
Definitions Course Introduction Classification Scheduling Exercises α | β | γ Classification Scheme Complexity Hierarchy Schedules Job Characteristics (2) α 1 α 2 | β 1 . . . β 13 | γ β 8 = brkdwn machine breakdowns β 9 = M j machine eligibility restrictions (if α = Pm ) β 10 = prmu permutation flow shop β 11 = block presence of blocking in flow shop (limited buffer) β 12 = nwt no-wait in flow shop (limited buffer) β 13 = recrc recirculation in job shop .::. 22
Definitions Course Introduction Classification Scheduling Exercises α | β | γ Classification Scheme Complexity Hierarchy Schedules Objective (always f ( C j ) ) α 1 α 2 | β 1 β 2 β 3 β 4 | γ Lateness L j = C j − d j Tardiness T j = max { C j − d j , 0 } = max { L j , 0 } Earliness E j = max { d j − C j , 0 } � 1 if C j > d j Unit penalty U j = 0 otherwise .::. 23
Definitions Course Introduction Classification Scheduling Exercises α | β | γ Classification Scheme Complexity Hierarchy Schedules Objective α 1 α 2 | β 1 β 2 β 3 β 4 | γ Makespan: Maximum completion C max = max { C 1 , . . . , C n } tends to max the use of machines Maximum lateness L max = max { L 1 , . . . , L n } Total completion time � C j (flow time) Total weighted completion time � w j · C j tends to min the av. num. of jobs in the system, ie, work in progress, or also the throughput time Discounted total weighted completion time � w j ( 1 − e − rC j ) Total weighted tardiness � w j · T j Weighted number of tardy jobs � w j U j All regular functions (nondecreasing in C 1 , . . . , C n ) except E i .::. 24
Definitions Course Introduction Classification Scheduling Exercises α | β | γ Classification Scheme Complexity Hierarchy Schedules Other Objectives α 1 α 2 | β 1 β 2 β 3 β 4 | γ Non regular objectives Min � w ′ j E j + � w ” j T j (just in time) Min waiting times Min set up times/costs Min transportation costs .::. 25
Definitions Course Introduction Classification Scheduling Exercises Exercises Complexity Hierarchy Schedules Gate Assignment at an Airport Airline terminal at a airport with dozes of gates and hundreds of arrivals each day. Gates and Airplanes have different characteristics Airplanes follow a certain schedule During the time the plane occupies a gate, it must go through a series of operations There is a scheduled departure time (due date) Performance measured in terms of on time departures. .::. 27
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